Maintenance of Chaos
Although chaos control may be very advantageous in many systems, it has been suggested that the pathological destruction of chaotic behavior may be implicated in heart failure and some types of brain seizures. For instance, some studies of heart rate variability suggest that losing complexity in the heart rate will increase the mortality rate of cardiac patients. Thus, some systems require chaos and/or complexity in order to function properly. Maintenance of chaos could also be useful in machine tool chattering applications for avoiding occurrences of low order periodic vibrational modes in the cutting bits during milling or cutting of parts. Other situations in which maintenance of chaos might be useful include the mixing of fluids and to apply maintenance of chaos to combustion systems in order to avoid flameout.
Experimental work by Schiff et al. demonstrated an ad hoc method for increasing the complexity of an in vitro rat brain hippocampal slice preparation. Recent theoretical and computational work by Yang et al. indicates that intermittent chaotic systems can be made to exhibit continuous chaotic behavior with no intermittent periodic episodes.
The main contribution of this work is a theoretical method for the maintenance of chaos, which is then implemented experimentally in a magnetoelastic ribbon demonstrating intermittency. This intermittency appears as chaos interspersed with long periodic episodes. The method presented here is readily applicable to experiments and relies only on experimentally measured quantities for its implementation.
Introduction
An ad hoc method for increasing the system complexity, which they termed anticontrol, was implemented by Schiff et al. Subsequent to this, Yang et al. proposed a method of anticontrol based on the observation that a mapbased system in a regime of transient chaos, such as that near a transition from periodicity into chaos, has special regions in its phase space that they term ěloss regionsî. If the system enters such a region, it immediately ceases its chaotic motion. Yang et al. identify these regions along with n preiterates of each loss region. If the system enters a preiterate, they apply a small perturbation to an accessible system parameter in order to interrupt the progression of the system toward a loss region. The perturbation places the system in a region of phase space that is neither a loss region nor a preiterate of one. This requires explicit knowledge of the map of the system and is accordingly difficult to accomplish experimentally.
A general anticontrol method was proposed that is readily
applicable to experiment and that relies only on experimentally
measured quantities for its implementation. Only the following
assumptions were made about the system: (1) the dynamics of the
system can be represented as an ndimensional nonlinear map (e.g.,
by a surface of section or a return map) such that points or
iterates on such a map are given by 
, where 
is some accessible system parameter; (2)
there is at least one specific region of the map (termed a ěloss
regionî) that lies on the attractor into which the iterates will
fall when making the transition from chaos to periodicity; and
(3) the structure of the map does not change significantly with
small changes 
in the control parameter
about some
initial value 
.
On the return map derived from a system, the locations of loss
regions were determined by observing immediate preiterates of
undesired points which correspond to a periodic orbit or a steady
state. Clusters of these preiterates were identified as the loss
regions. The extent of each loss region was determined by the
experimentally observed distribution of points in that region.
The time evolution of each region could be traced back through 
preiterates, as
desired.
In a fashion similar to the OGY chaos control method, 
was changed
slightly; one then observed the resulting change in each loss
regionís location and estimated the local shift of the attractor
for each
loss region with respect to a change in 
as follows:
. (4.1)
As an approximation, 
was taken to be constant for all loss regions on the
attractor for sufficiently small parameter changes 
(otherwise
calculation of 
for each loss region would be required). This was
not strictly necessary in order to implement the method, but was
simply a convenience that is approximately true for many systems
(including our magnetoelastic ribbon) for small 
ís.
Anticontrol can be applied once the system has entered the 
preiterate of
the loss region. Since the map is constructed as a return map (or
a delay coordinate embedding) with 
versus 
, the y-coordinate of the 
point becomes
the x-coordinate of the 
point. Since known values include the x-coordinate
of the next point and the size of the region that this 
point would
normally fall into, a minimum distance is calculated to move the
attractor so that this next point falls outside of that region.
This distance 
is translated into the appropriate parameter change 
by
, (4.2)
where the direction of the motion is along 
.
If each of the 
preiterates of the loss region is circumscribed by a
circle of radius 
, the worst case scenario gives 
, where it is
understood that the 
point falls into the 
preiterate region. This is the maximum
perturbation needed to achieve anticontrol and it guarantees that
the next point will fall outside the 
preiterate
region by moving the point one full diameter of the circle
surrounding the loss region. Yet, this worst case can be improved
upon. With a return map, the x-coordinate of the next point is
known. Because there is a choice of whether to apply the
perturbation in either the positive or the negative 
direction, the
sign of the perturbation can be selected to move the next point
to the left if this x-coordinate is in the left half of the
preiterate region and vice versa. Thus, the minimum distance to
move this x-coordinate is reduced to 
and consequently
, resulting
in a 50% reduction in the strength of the perturbation.
Additionally, if the shape of the preiterate region of
interest is approximately linear (line-like) and its slope is
perpendicular to 
, then 
is at most 
and may even approach the thickness of this linear
segment (
).
Thus, while not necessary to achieve anticontrol, a detailed
knowledge of the shape of the loss region and its
preiterates can further reduce the size of the perturbation
required to achieve anticontrol.
The experimental system consists of a gravitationally buckled
magnetoelastic ribbon driven parametrically by a sinusoidally
varying magnetic field. An ac magnetic field of amplitude 
and frequency 
added to a dc
field of amplitude 
were applied such that 
. The values were chosen to be 
, 
and 
in order to
establish the system in a state of intermittent chaos. During the
intermittent chaos, the system would switch between a periodic
attractor and a chaotic attractor without any outside
intervention. A return map was constructed by measuring the
position, 
,
of a point on the ribbon once every driving period and then
plotting the current position 
versus 
, where 
is the delay. Here, where the data was
strobed at the driving frequency, it resulted in 
.
The loss region and its preiterates were identified on the
return map. The loss region (
) is denoted by the circle immediately to
the left of the diagonal. Its first preiterate (
) lies to the
right of the diagonal. The other circles denote earlier
preiterates (
). A point that entered any of these regions
would eventually go to the region 
. Once there, the point would proceed into
the loss region 
on the next iterate. Then the system became
periodic. This appears as a cluster of points (
) on the diagonal
of the map. The points that enter the preiterate regions mediate
the intermittent transition from chaos to periodicity. During
anticontrol, a perturbation was applied when an orbit entered the
region 
, so
that the next orbit would fall outside of 
.
The extent of the 
preiterate region was determined by observing the
set of points that, after 
iterations, fall into the loss region as
well as neighboring points that do not fall into the loss region
after 
iterations. The boundary of the loss region lies between these
points. The 
vector was determined by changing 
by 
. The two loss regions were very close to
the cluster denoting the periodic orbit. Rarely during
anticontrol the orbit was kicked into the period one region. This
required the implementation of anticontrol on the succeeding
iteration for the periodic region as well, in order to safeguard
against the system remaining there. Of course when this happened,
a somewhat larger perturbation would be required to move the
system away from this periodic orbit. A value of 0.0237 Oe
was adequate to control this problem in this experiment. A more
elegant but computationally difficult solution would be to choose
the original perturbation so that it avoided all of the loss
regions and preiterates.
In summary, a general method for the anticontrol of chaotic
systems has been presented which is straightforward to implement
and only need be applied infrequently to keep a system chaotic.
The method required the identification of the loss region
location in the map and the locations of the preiterate regions.
During maintenance of chaos the perturbation was applied to the
system whenever the orbit entered the preiterate regions, which
indicated the sequencing toward the loss region. The size of the
perturbation was determined by a distance needed to move the next
iterate so that the orbit would no longer be within the loss
region or its preiterate region. In addition, several methods of
reducing the perturbation based on the nature of return maps and
on the geometry of the experimentally measured attractor have
been discussed. Maintenance of chaos was demonstrated in the
magnetoelastic ribbon experiment that was arranged to exhibit
intermittency. During this intermittency the system was switching
between chaos and period1 at irregular intervals. The undesired
period1 episodes were eliminated by applying occasional
perturbations during maintenance of chaos. The importance of this
technique is that it can be applied to experiments with no apriori
knowlege of the systemís dynamics. Potential applications of
this technique include supression of eipileptic seizures,
reduction of tool chatter and the elimination of flameout in
combustion processes.